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%I #20 Jan 30 2020 21:29:17
%S 1,1,2,2,6,10,26,54,134,306,754,1806,4478,11018,27578,69014,174358,
%T 441506,1124674,2873182,7370926,18962490,48939530,126625126,328475334,
%U 853983058,2225023890,5808480046,15191255262,39798134122,104431095898,274439510838,722232378998,1903192922626,5021490842338,13264643729406
%N Expansion of (1-x+2*x^3-sqrt(1-2*x-3*x^2+4*x^3-4*x^4))/(2*x^2).
%H S. Kitaev and J. Remmel, <a href="https://arxiv.org/abs/1305.6970">The 1-box pattern on pattern avoiding permutations</a>, arXiv preprint arXiv:1305.6970 [math.CO], 2013. See Th. 8.
%F D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) -3*(n-1)*a(n-2) +2*(2*n-5)*a(n-3) -4*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jan 24 2018
%F a(n) = Sum_{i=0..n+1} Sum_{k=0..n-i} C(i,k)*C(k+i,i)*Sum_{j=0..(-n+3*k+i+2)/2} C(k+1,j)*C(k-j+1,n-2*k+j-i-1)*(-1)^(-n+k+i))/(k+1), a(1)=1. - _Vladimir Kruchinin_, May 07 2018
%t CoefficientList[Series[(1 - x + 2 x^3 - Sqrt[1 - 2 x - 3 x^2 + 4 x^3 - 4 x^4]) / (2 x^2), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 30 2019 *)
%o (Maxima)
%o a(n):=if n=1 then 1 else sum(sum((binomial(i,k)*binomial(k+i,i)*(sum(binomial(k+1,j)*binomial(k-j+1,n-2*k+j-i-1),j,0,(-n+3*k+i+2)/2))*(-1)^(-n+k+i))/(k+1),k,0,n-i),i,0,n+1); /* _Vladimir Kruchinin_, May 07 2018 */
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Oct 01 2013
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